Most existing industrial products designed for multivariable model predictive control (MPC) employ linear step-response models or finite impulse response (FIR) models. These approaches result in over-parameterization of the models (Qin and Badgwell, 1996). For example, the dynamics of a first order single input/single output SISO process which can be represented with only three parameters (gain, time constant and dead-time) in a parametric form typically require from 30 to 120 coefficients to describe in a step-response or FIR model. This over-parameterization problem is exacerbated for non-linear models since standard non-parametric approaches, such as Volterra series, lead to an exponential growth in the number of parameters to be identified. An alternative way to overcome these problems for non-linear systems is the use of parametric models such as input-output Nonlinear Auto-Regressive with eXogenous inputs (NARX). Though NARX models are found in many case-studies, a problem with NARX models using feed forward neural networks is that they offer only short-term predictions (Su, et al, 1992). MPC controllers require dynamic models capable of providing long-term predictions. Recurrent neural networks with internal or external feedback connections provide a better solution to the long-term prediction problem, but training such models is very difficult.
The approach described in (Graettinger, et al, 1994) and (Zhao, et al, 1997) provides a partial solution to this dilemma. The process model is identified based on a set of decoupled first order dynamic filters. The use of a group of first order dynamic filters in the input layer of the model enhances noise immunity by eliminating the output interaction found in NARX models. This structure circumvents the difficulty of training a recurrent neural network, while achieving good long-term predictions. However, using this structure to identify process responses that are second order or higher can result in over sensitive coefficients and in undesirable interactions between the first order filters. In addition, this approach usually results in an oversized model structure in order to achieve sufficient accuracy, and the model is not capable of modeling complex dynamics such as oscillatory effects. In the single input variable case, this first order structure is a special case of a more general nonlinear modeling approach described (Sentoni et al., 1996) that is proven to be able to approximate any discrete, causal, time invariant, nonlinear SISO process with fading memory. In this approach a Laguerre expansion creates a cascaded configuration of a low pass and several identical band pass first order filters. One of the problems of this approach is that may it require an excessively large degree of expansion to obtain sufficient accuracy. Also, it has not been known until now how to extend this methodology in a practical way to a multi-input system.
This invention addresses many essential issues for practical non-linear multivariable MPC. It provides the capability to accurately identify non-linear dynamic processes with a structure that    has close to minimum parameterization    can be practically identified with sufficient accuracy    makes good physical sense and allows incorporation of process knowledge    can be proven to identify a large class of practical processes    can provide the necessary information for process control